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Math Help - Boolean ring

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    Boolean ring

    Prove that in a Boole ring with elements there exists s.t. for all distinct .
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    Quote Originally Posted by petter View Post
    Prove that in a Boolean ring with elements there exists s.t. for all distinct .
    since the ring R is Boolean, it is unitary, commutative and has no non-zero nilpotent element. hence R is the direct sum of n copies of \mathbb{F}_2. now for any 1 \leq i \leq n let a_i=(x_{i1}, \cdots , x_{in}) \in R,

    where x_{ii}=1 and x_{ik}=0, \ \forall k \neq i. obviously a_ia_j=0, \ \forall i \neq j. \ \ \Box
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